week ending PRL 105, 267204 (2010) PHYSICAL REVIEW LETTERS 31 DECEMBER 2010 Effective Spin Model for the Spin-Liquid Phase of the Hubbard Model on the Triangular Lattice Hong-Yu Yang,1 Andreas M. La¨uchli,2 Fre´de´ric Mila,3 and Kai Phillip Schmidt1 1Lehrstuhl fu¨r Theoretische Physik I, Otto-Hahn-Straße 4, TU Dortmund, D-44221 Dortmund, Germany 2Max Planck Institut fu¨r Physik komplexer Systeme, D-01187 Dresden, Germany 3Institute of Theoretical Physics, Ecole Polytechnique Fe´de´rale de Lausanne, CH-1015 Lausanne, Switzerland (Received 29 June 2010; revised manuscript received 23 October 2010; published 29 December 2010) We show that the spin-liquid phase of the half-filled Hubbard model on the triangular lattice can be described by a pure spin model. This is based on a high-order strong coupling expansion (up to order 12) using perturbative continuous unitary transformations. The resulting spin model is consistent with a transition from three-sublattice long-range magnetic order to an insulating spin-liquid phase, and with a jump of the double occupancy at the transition. Exact diagonalizations of both models show that the effective spin model is quantitatively accurate well into the spin-liquid phase, and a comparison with the Gutzwiller projected Fermi sea suggests a gapless spectrum and a spinon Fermi surface. DOI: 10.1103/PhysRevLett.105.267204 PACS numbers: 75.10.Jm, 05.30.Rt, 75.10.Kt Although the Hubbard model has been one of the central strong coupling limit using perturbative continuous unitary paradigms in the field of strongly correlated systems for transformations (PCUTs), and by showing that it gives a about five decades, new aspects of its extremely rich phase qualitative and quantitative account of the transition from diagram are regularly unveiled. Even at half-filling, the the three-sublattice magnetic order to the SL state. popular wisdom according to which the model has only The starting point is the single-band Hubbard model on two phases, a metallic one at weak coupling and an insu- the triangular lattice defined by the Hamiltonian lating one at strong coupling separated by a first order X X H ¼ H þH ¼ U n n Àt ðcy c þH:c:Þ transition [1], has been recently challenged. This goes U t i" i# i j (1) i back to the work of Morita et al. [2] on the triangular hi;ji; lattice which revealed the presence of a nonmagnetic in- where the sum over i runs over the sites of a triangular sulating phase close to the metal-insulator transition using lattice, and the sum over hi; ji over pairs of nearest neigh- a path integral renormalization group. Further evidence in bors. To derive an effective low-energy Hamiltonian, favor of a phase transition inside the insulating phase has we use PCUTs [10–13] about the localized limit treating been reported using a variety of theoretical tools [3–6]. the hopping term as a perturbation. The kinetic part can be More recently, an intermediate spin-liquid (SL) phase has written as Ht ¼ tðT0 þ T1 þ TÀ1Þ where Tm changes the also been identified on the honeycomb lattice using quan- number of doubly occupied sites by m. The PCUT method tum Monte Carlo simulations [7]. provides order by order in t=U an effective Hamiltonian The precise nature of the SL phase of the Hubbard model Heff with the property ½HU;Heff¼0; i.e., Heff is block on the triangular lattice is of direct experimental relevance diagonal in the number of doubly occupied sites. The low- for the 2D organic salt -ðBEDT-TTFÞ2Cu2ðCNÞ3 [8]. energy physics at half filling is expected to be contained in 0 As such, it has already attracted a lot of attention, but the block with no doubly occupied site Heff, which can be fundamental questions such as the appropriate low-energy expressed as a spin-1=2 model of the form effective theory remain unanswered. Since the phase is X 0 ~ ~ insulating, an effective model where charge fluctuations Heff ¼ constþ Jn~ ðSr~ ÁSr~þn~ Þ r;~ n~ are treated as virtual excitations should be possible. X One step forward in this direction has been taken by n~3 ~ ~ ~ ~ þ J ðSr~ ÁSr~þn~ ÞðSr~þn~ ÁSr~þn~ Þþ... (2) n~1;n~2 1 2 3 Motrunich [9], who proposed to describe the SL phase r;~ n~1;n~2;n~3 with four-spin interactions. However, whether a description in terms of a pure spin model is possible is far from obvious, where r~ and r~ þ n~ denote sites on the triangular lattice. in particular since there seems to be a jump in the double All remaining terms can be written in a similar way as ~ ~ occupancy at the transition from the three-sublattice Ne´el products of Sr~ Á Sr~0 due to the SU(2) symmetry of the phase to the SL [2,6]. Hubbard model. In this Letter, we show that the correct low-energy theory The PCUT provides the magnetic exchange couplings as of both insulating phases and, in particular, of the SL phase, series expansions in t=U. Since the spectrum of the is indeed a pure spin model. This has been achieved by Hubbard model is symmetric at half filling under the deriving an effective spin model to very high order about the exchange t $t, only even order contributions are 0031-9007=10=105(26)=267204(4) 267204-1 Ó 2010 The American Physical Society week ending PRL 105, 267204 (2010) PHYSICAL REVIEW LETTERS 31 DECEMBER 2010 present [14]. We have determined all two-spin, four-spin which indicates the presence of a first order transition in and six-spin interactions up to order 12. The obtained the thermodynamic limit. Let us first investigate to which series are valid in the thermodynamic limit. To this end extent the effective spin model provides an accurate de- we have fully exploited for the first time in a PCUT scription in this parameter range. The proof that this is so calculation the linked cluster theorem by using graph relies on the following four observations. theory. At order 12 there are 1336 topologically different Convergence of the series.—As usual, we use Pade´ ap- graphs. The major complication for the determination of proximants to extrapolate the series, and we consider that higher orders comes from the large number of different the series are well converged as long as different Pade spin operators, which requires us to calculate all possible approximants give consistent results. Here this is definitely matrix elements for each graph. true up to t=U ’ 0:125 (U=t ’ 8), i.e., well beyond the level The most relevant spin couplings are shown in Fig. 1 as a crossing [see inset of Fig. 1]. For all couplings, the absolute function of t=U. As already pointed out in Ref. [9], the change between order 10 and 12 is below 10À4 (10À3)for most important correction to the nearest-neighbor t=U ¼ 0:1 (t=U ¼ 0:125). In fact, clear indications of di- Heisenberg exchange at order 4 are the four-spin interac- vergence only appear above t=U ’ 0:15, and this is proba- tions on a diamond plaquette. High-order contributions bly related to the metal-insulator transition. result in more complex multispin interactions with larger Comparison of GS energy with Hubbard model.—In spatial extension but also lead to renormalizations of al- view of the anomaly observed at the transition by other ready existing couplings. The latter effect causes, for ex- approaches [2,6], it is legitimate to ask whether the GS ample, a sizable splitting of the two four-spin interactions remains in the sector with no doubly occupied site. To shown in Fig. 1, which have the same absolute value at address this point, we have used ED of finite clusters to order 4. Processes involving sites far apart have small calculate the GS energy of both the Hubbard model and the amplitudes, and multispin interactions between sites not effective spin model as a function of U=t [see Figs. 2(a) far from each other are clearly the dominant corrections to and 2(b)] up to 21 sites for the Hubbard model, and up to the spin operators at order 4. 36 sites for the effective spin model. We have included in At large U=t, nearest-neighbor exchange dominates, and our calculation all spin operators appearing up to order 6. the system is expected to develop three-sublattice long- Additionally, we have omitted for a given ratio t=U all range order [15–17]. Using exact diagonalizations (ED) of terms whose coupling was smaller than 10À6. When em- clusters up to 36 sites, we have investigated how the bedded on our largest 36-site cluster, this corresponds to properties of the model evolve upon reducing U=t. The 16 000 distinct terms in total. The resulting total energies most striking feature is a level crossing in the ground state per site are displayed in Fig. 2(a). The energy per site of the (GS) at U=t ’ 10 [see Fig. 2]. This level crossing is ob- effective spin model depends very little on the size, and for served for all cluster sizes at essentially the same value, the 21-site cluster, the energies of the Hubbard model and of the effective spin model are in very good agreement on -0.3 0.002 (a) (b) 0.001 -0.4 J1,0 N=12, Spin model 0 -0.5 N=21, Spin model N=27, Spin model N=36, Spin model -0.001 4sp N=21, Hubbard -J N=36, PIRG Ref.